A probability paradox?

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understandinguncertainty.org was produced by the Winton programme for the public understanding of risk based in the Statistical Laboratory in the University of Cambridge. The aim was to help improve the way that uncertainty and risk are discussed in society, and show how probability and statistics can be both useful and entertaining.

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I recently tweeted a link to this problem drawn on a blackboard, which got a lot of retweets.

Multiple Choice: If you choose an answer to this question at random, what is the chance you will be correct? A) 25% B) 50% C) 60% D) 25%

This is a fun question whose paradoxical, self-referential nature quickly reveals itself – A) seems to be fine until one realizes the D) option is also 25%.

A quick search reveals hundreds of discussion contributions of this problem, for example here and here and from a year ago. People often appear very confident that their answer is the only possible solution.

I am no logician and so unqualified to place this within the grand structures of mathematical paradoxes. I have not waded through all the discussions and so there may be something I have missed, but in among all the arguments there seem to be four conclusions that could be considered as 'correct'. These are my personal comments:

1) There can be no solution, since the ambiguity of ‘correct’ makes the question ill-posed.

It's true the question is ambiguous, but this still seems a bit of a cop-out.

2) There is no solution.

This seems to take this interpretation of the question.

Which answer (or set of answers) of “p%”, is such that the statement ‘the probability of picking such an answer is p%’ is true?

Then this appears to be a well-posed question, but there is no solution.

3) 0%.

Consider a different interpretation of the question.

Is there a p%, such that the statement ‘the probability of picking an answer “p%” is p%’ is true?

Then this appears a well-posed question and has the solution p = 0, even though this is not one of the answers. Of course if answer C) were changed to “0%” (as it is in this 2007 version of the question ), then this would also have no solution.

4) We can produce any answer we want by changing the probability distribution for the choice.

Why should ‘random’ mean an equally likely chance of picking the 4 answers? If we, say, assume the probabilities of choosing (A) (B) (C) (D) to be (10%, 20%, 60%, 10%) then the answer to either formulation (2) and (3) is now “60%”. But if we make the distribution (12.5%, 15%, 60%, 12.5%) then we seem to back to square one again, since there is now both a 25% chance of picking “25%”, and a 60% chance of picking “60%”.

I like conclusion 3) best, ie 0%.

Maybe the main lesson is: ambiguity and paradox are often the basis for a good joke.



I consider it a feedback loop with no stopping citerion: 'Inputting' an answer of 50% on the first iteration makes the next iteration return an answer of 25%, which in turn puts the NEXT iteration at 50% etc. etc. One might as well have asked "Is the statement that "my previous statement was true" true or false?"

But what if none of the answers is correct?

If the answer is the probability of getting it right, it keeps going on and on so you never know if it's right and there is no way to fid out because you have to determine whether it's correct over and over and over again.

David, I see no paradox. The answer is surely 0%. I give my reasoning at http://djmarsay.wordpress.com/2012/02/01/uncertainty-puzzle/ . I also give a variant that would be a probability if you thought that everything that looked like a well-formed question was, and that all probabilties are numeric. I think that it is well-formed, so it must be an example of a non-numeric probability. But you may want to ask a logician.

I meant ... would be a paradox if ..., i.e. some things that linguistically seem to be probabilities aren't.

Dave, your answer is clearly wrong. 0% is not a valid answer to a multiple choice question.

Let me give you a simple example to help you understand your error.

Multiple Choice: If you flip a fair coin, what is the probability of getting heads?
A) 75%
B) 25%
C) 100%
D) All of the above
E) None of the above

The answer is E. That's how multiple choice questions are answered. If you write 50% below the answers on the test paper, you would be marked wrong. Writing a number is a NOT a valid answer to a multiple choice question even if the question is "what is the probability".

If you still think that the answer to the paradox is 0%, then you have shown a basic failure in your ability to follow instructions.

Let me give you another example:

Multiple Choice: What is 1+1?
A) 22
B) 11
C) 0

What is the answer? There is no correct answer! Would this question make you start making up nonsense answers like you did in the Paradox question? Or would you just admit the question is stupid and has no answer? Why is it that people can't grasp that the paradox question simply has no valid answer? What is it about the question that causes people keep making up nonsense answers and try to argue in support of the nonsense?

It's like me saying the answer to this last question is A), and giving the reason that it is because the teacher just accidently made a typo when typing up the question and obviously meant to type a single 2! The question as written is clear, and the question as written, simply has no valid answer. End of story. Nothing else to discuss.

I think that if you look at the question. It asks if you randomly choose, what are the chances YOU would be correct? Well then I say it is up to me if the answer is right. So there for it is right 100% of the time. Now then 100% is not one of the A B C D answers... But does it really tell me I have to pick from those 4?

Basically, none of the answers. The probability of getting it right is 33.33333% because there are actually three percentages, since there is twice the number 25, so it's one out of three to get it right. That is my humble opinion.

I love this question because it had so many layers. First thought was of course, what they said in school. "If you don't know the answer, you have a one in four chance if you guess." But then you see the other 25%... OK now it's 50%... but wait now there is a is a 0%... but that's not one of the answers ether. Take the answers out, and answer THIS question.

Multiple Choice: If you choose an answer to this question at random, what is the chance you will be correct? A B C D

or what if the answers were A)Blue B)Yellow C)Red D)Blue

This answer implies that there is another question. But the question states "this question"

So if I choose Yellow, Red, or ether Blue, I'm right... What if the implied question was whats my favorite color?

The question doesn't imply that any of the answers are necessarily correct. So in that sense the actual answer to the question isn't tied to the multiple choice options. An equivalent question would be: What are the chances of answering this question "what is the capital of Spain" correct. A) Paris B) London C) Tokyo D) Wellington. You would have a zero percent chance of answering the question correctly. Every option in the probability example is also wrong, therefore the answer is zero percent chance of answering correctly.

You always have an answer to the question :
1. How many answers are right from a set of four possible answers?
It is either 0, 1, 2, 3 or 4.

You also always have an answer to the question :
2. What is the probability of randomly picking the right answer from a set of four alternatives, given a fixed number of right answers?
It is either 0%, 25%, 50%, 75%, or 100%.

In this case they’re sort of asking you both questions simultaneously, since the content of the possible answers should be fixing and allowing you to discover both
a. the number of right answers
b. the right answer

Once you give an answer to either 1 or 2 you immediately get an answer to the other. In this case none of those answers is actually fixed, you should discover both at the same time: hence the (apparent) problem.
The trouble is that there are very there are some (actually very strict) constrains regarding the possible relation between a and b that are not being respected by the given possible answers: hence the incoherence.
Notice that for someone to be able to pick an answer correctly the following conditions must be met:
I) There is one correct answer ←→ There is one and only one option = 25%
II) There are two correct answers ←→ There should be two and only two options = 50%
III) There are three correct answers ←→ There should be three and only three options = 75%
IV) There are four correct answers ←→ There should be four and only four options = 100%
once you don’t respect this conditions you have an incompatibility between picking the right answer and picking the number of right answers, and so there is no way to answer the question and you should not be bogged by the appearance that there should be such an answer.

Here's where I got to...

Suppose one of the three percentages is correct. Then:

P(correct|right(25)) = 1/2
P(correct|right(50)) = 1/4
P(correct|right(65)) = 1/4

P(correct & right(25)) = 1/2 * 1/3 = 1/6
P(correct & right(50)) = 1/4 * 1/3 = 1/12
P(correct & right(65)) = 1/4 * 1/3 = 1/12

So P(correct) = 4/12 = 1/3

None of the a,b,c,d

Try again:

Four possible answers: a, b, c, d

Correct answer just 1/4

Again: But if we look at the content of the answers, then two of them are 1/4, so it's 1/2

BUT i think it should be 50 %....... cause answer -A and -D can't be occur , it doesn't make sense . BECAUSE , if you choose A , D also will be correct !!! , so the probability would be 50 % , , but the answer of A , D is 25 % , which means answer A , D will never be correct. !!!!! so the options which works will only be B and C , so it s only 1/2 chance to get the correct answer .... i am not sure , please suggest !!!!!

The answer ------------------- ditch diggers ------------------- -50%- -------------------- nervous -25%- - --------------------- Employable -25%+log(1+e^.5-e^-.5)/2- -------------------- ------------ Creators- ---------- 100%---------

A friend posted this on my FB account and I have spent the last couple of hours trying to figure it out. After coming up with a thousand different answers I jumped on here and saw I'd missed the point entirely. I looked at it like this:

The question says "If you pick an answer to this question at random..." I think that's loaded with implications...

It never says I have to choose an answer from the ones provided, just that I have to provide "an answer at random". It's not a 1 in 4 probability because I can give any answer I choose. "purple monkey dishwasher" is a valid answer. 17 is a valid answer. The question doesn't limit me to the four choices provided.

Pus, there are no values (numeric or otherwise) provided in the question. How is the answer calculated? Someone above asked "what if the question is 'what is the capital of Spain?'" Well then, there is an answer and only one correct answer. But what if the question was "what's your opinion of toasted cheese sandwiches?" What I mean is who decides whether the answer I provide is correct or not if there are no ways of calculating the answer in the first place? That implies to me that the answer is subject to evaluation and gradation by the person asking the question. I guess that would mean I can be anywhere from completely incorrect (zero%) or completely correct (100%). It's "correctness" is not a cold-hard right or wrong.

Finally, I thought "how can I know my chances of picking a correct answer if I don't know the question?". No question is provided. To which my friend replied "it IS the question". again, the wording is ambiguous. "If you provide an answer to THIS question". That means the question is self-referential. Wouldn't that mean it becomes an infinite regression, a logic-loop?

In the end I jumped online to find "the answer" only to discover I have missed the point entirely. I guess that's why my education ended in high school :) It's comforting to know that the university-educated aren't having much more luck than I am.

Oh, and I also felt that the question was attempting to be confusing by providing percentages as the answers. I started thinking it would be best to substitute the values provided with symbols ie:

A) 25% (substitute "orange")
B) 50% ("apple")
C) 60% ("banana")
D) 25% ("orange")

as long as you still have 3 different answers out of four possible choices it removes the confusion but still leaves the same probability of getting the "right" answer; assuming you are approaching it as a probability question.

My head hurts.

Sorry, just re-read the above posts and saw a few points I made had already been pointed out... apologies.

now change the possible answers to: a)25% b)50% c)75% d)50%

E) 20% Now, where is my medal?

actually no it isn't I spent 5 minutes working on this problem or what it was meant to be a meam but anyway. the question does not specify that the a b or c is showing the answer they could be





they could be that and would not change the question it's not a paradox and

there a

50% chance to land on dog

25% chance to land on cat

25% chance to land on box

dog, cat, box each one could be the correct answer so if you look at the probability

if box is correct then there a 25 chance you get the right answer on random

if cat is correct then there a 25 chance you get the right answer on random

if dog is correct then there a 50 chance you get the right answer on random

with this now you can use average to figure out the chance to land on the correct answer



so the answer is 33.3333333333333333333% on landing on the correct answer. this is the chance if we don't know which between a b or c is correct and d but d is the same as a.

Hi, I don't think it's a paradox, it's just sampling of states/events that you can assume not to be dependent, because you have no knowledge they are. The answer will be the sum of the distribution of probability*value.

Our testing is set up as follows: For every answer we pick, we can check the probability of the outcome by assuming we do an independent pick to test the described outcome.
3 of the events have a non-zero probability.

There are 4 events in this sampling, distribution sampling a, b,c, or d:
a: you select 0.25 as an answer. This is true with a probability of 0.50 (in case you would take an independent sample where you select answer b) P*value=0.125
b: you select 0.50 as an answer. This is true with a probability of 0.50 (in case you would take an independent sample where you select answer a or d), P*value=0.25
c: you select 0.60 as an answer. This is true with a probability of 0, P*value=0 (none of the independent sampling would result in this answer)
d: you select 0.25 as an answer. This is true with a probability of 0.50 (when you assume you take an independent sample where you select answer b), P*value=0.125

The sum of the probability*value for this system is therefore 0.125+0.25+0+0.125=0.50

The correct answer is then b: 50%.